The sum of two numbers is $78$, and their difference is $14$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 78}$ ${x-y = 14}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 92 $ $ x = \dfrac{92}{2} $ ${x = 46}$ Now that you know ${x = 46}$ , plug it back into $ {x+y = 78}$ to find $y$ ${(46)}{ + y = 78}$ ${y = 32}$ You can also plug ${x = 46}$ into $ {x-y = 14}$ and get the same answer for $y$ ${(46)}{ - y = 14}$ ${y = 32}$ Therefore, the larger number is $46$, and the smaller number is $32$.